TY - JOUR

T1 - An example of a convex body without symmetric projections

AU - Gluskin, E. D.

AU - Litvak, A. E.

AU - Tomczak-Jaegermann, N.

PY - 2001

Y1 - 2001

N2 - Many crucial results of the asymptotic theory of symmetric convex bodies were extended to the non-symmetric case in recent years. That led to the conjecture that for every n-dimensional convex body K there exists a projection P of rank k, proportional to n, such that PK is almost symmetric. We prove that the conjecture does not hold. More precisely, we construct an n-dimensional convex body K such that for every k > C√n ln n and every projection P of rank k, the body PK is very far from being symmetric. In particular, our example shows that one cannot expect a formal argument extending the "symmetric" theory to the general case.

AB - Many crucial results of the asymptotic theory of symmetric convex bodies were extended to the non-symmetric case in recent years. That led to the conjecture that for every n-dimensional convex body K there exists a projection P of rank k, proportional to n, such that PK is almost symmetric. We prove that the conjecture does not hold. More precisely, we construct an n-dimensional convex body K such that for every k > C√n ln n and every projection P of rank k, the body PK is very far from being symmetric. In particular, our example shows that one cannot expect a formal argument extending the "symmetric" theory to the general case.

UR - http://www.scopus.com/inward/record.url?scp=0035649455&partnerID=8YFLogxK

U2 - 10.1007/BF02772622

DO - 10.1007/BF02772622

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AN - SCOPUS:0035649455

SN - 0021-2172

VL - 124

SP - 267

EP - 277

JO - Israel Journal of Mathematics

JF - Israel Journal of Mathematics

ER -